Partition Volume Variability in Digital Polymerase Chain Reaction

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Partition Volume Variability in Digital Polymerase Chain Reaction Methods: Polydispersity Causes Bias but Can Improve Precision Joel Tellinghuisen Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b03139 • Publication Date (Web): 15 Nov 2016 Downloaded from http://pubs.acs.org on November 19, 2016

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Analytical Chemistry

Partition Volume Variability in Digital Polymerase Chain Reaction Methods: Polydispersity Causes Bias but Can Improve Precision

Joel Tellinghuisen,* Department of Chemistry, Vanderbilt University Nashville, Tennessee, USA 37235

Author:

Joel Tellinghuisen* Department of Chemistry Vanderbilt University Nashville, TN 37235

Phone:

(615) 322-4873

FAX:

(615) 343-1234

Email:

[email protected]

ACS Paragon Plus Environment


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-2ABSTRACT The role of partition volume variability, or polydispersity, in digital polymerase chain reaction methods is examined through formal considerations and Monte Carlo simulations. Contrary to intuition, polydispersity causes little precision loss for low average copy number per partition µ and can actually improve precision when µ exceeds ~4. It does this by negatively biasing the estimates of µ, thus increasing the number of negative (null) partitions N0. In keeping with binomial statistics, this increases the relative precision of N0 and hence of the biased estimate m of µ. Below µ = 1, the precision loss and the bias are both small enough to be negligible for many applications. For higher µ the bias becomes more important than the imprecision, making accuracy dependent on knowledge of the partition volume distribution function. This information can be gained with optical microscopy or through calibration with reference materials.

Key Words:

dPCR, data analysis, partition volume dispersion, Monte Carlo


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INTRODUCTION Although the principles behind both digital and analog (real-time quantitative) polymerase chain reaction (PCR) analytical methods have been known for a quarter of a century,1,2 it is only in recent years that the digital version has come into prominence, as techniques for producing, reading, and analyzing results from large numbers of partitions have evolved.3-14 The method is based on distributing the genetic material over the partitions in such a way that, after PCR amplification and interrogation, measureable fractions of the droplets will be found to be negative (no DNA) and positive. Then, if the partitions all have the same volume, the Poisson probability of a negative (or null) is e−µ, where µ is the average number of copies per partition. This is also the predicted fraction of all partitions that are nulls, and µ can be estimated from

µ = ln(N/N0),

(1)

where N0 is the number of nulls and N the total number of partitions. The concentration is then determined as λ = µ/v, with v the partition volume. In keeping with this treatment, much experimental effort has been directed toward producing partitions of constant volume (monodisperse) and well-determined average v. Efforts to determine the significance of polydispersity have indicated typical standard deviations of several percent in the volume8,14 or radius9 of droplet systems, but have provided little information about the distribution functions (illustrated as Gaussian in the radius in Ref. 9, but with limited statistical support). At least two studies have examined the effects of polydispersity through modeling and, consistent with simple expectations, have found that increasing volume dispersion leads to increasing loss of precision in estimating λ.11,13 However, I have found that such precision losses have little practical significance, and in fact under some circumstances polydispersity can actually increase precision. In this work I use simple models and Monte Carlo simulations to explore this surprising finding. The results show that polydispersity leads to bias that increases with µ, and to increased precision for large µ. The bias is negative, and so the reason for the increased precision in this µ regime is simple: the low-biased estimate (m) of µ corresponds to larger N0, hence larger relative



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precision in N0 and in m. Of course, good information about the v distribution function is needed to determine the bias and correct m to µ. Some of the precision gain is lost in applying this correction, but enough is preserved to make the overall precision better when µ > ~4. For µ < 1, both effects diminish in significance. Thus, experimental methods that can increase throughput and decrease costs by sacrificing monodispersity can be expected to perform well at small µ. Knowledge of the average v is still important, but can be determined, in principle, through calibration with reliable reference materials.15,16 COMPUTATIONAL METHODS Statistical considerations For monodisperse partitions, the numbers of nulls and hits follow binomial statistics, from which the estimated standard deviation (SD) in N0 is σN0 = [N0 (1−N0/N)]1/2, whereupon the relative SD (RSD) is

σN0

1 eµ−1 1 = [N − N ]1/2 = [ N ]1/2 , 0 N0

(2)

where eq 1 is used to obtain the final expression on the right. From eq 1 the SD in µ is also the RSD in N0. This quantity decreases with decreasing µ, but for small µ, the RSD in µ (σµ/µ) becomes (µN)−1/2, which increases with decreasing µ. The RSD in µ increases also at large µ, from the dominance of the exponential. It has its minimum near µ = 1.59.11 Next, suppose we have two volumes that average 1, for example, N/2 partitions each of v = 0.9 and 1.1. The fractions of nulls are, respectively, e−0.9µ and e−1.1µ, and the estimated µ becomes m = ln(N/N0) = ln2 − ln(e−0.9µ + e−1.1µ) ,

(3)

with variance σm2 = (em−1)/N (eq 2). m is now a biased estimator of µ, e.g. 1.98 for µ = 2. The RSD in m is smaller than that for µ for monodisperse v when µ > 1.6. Correcting for the bias offsets some of this precision gain, but for sufficiently large m, we find net precision gain (see below). If the partitions have variable volume with distribution function f(v), the average probability of a null over the partition is


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-5〈 P0 〉 =

v

∫ 0 max e−Μ(v) f (v)dv .

(4)

where Μ(v) represents the v-dependent µ. If f(v) is chosen to have average v = 1, then Μ(v) = µ v. Thus, for example, if f(v) is the uniform distribution over range 1 ± s (s < 1) (and for any average v = v0 ± s v0), we obtain

〈P0〉 = e- µ sinh(µs) , muni = −ln〈P0〉 ≈ µ −

and

µ 2 s2 6

(5) ,

(6)

where the approximation includes just the lead term in (µ s). For a normal distribution having RSD s, we obtain m norm ≈ µ −

µ 2 s2

,

2

(7)

where the approximation comes from extending the lower limit of integration in eq 4 to −∞, which is approximately valid for s < 0.25. (Numerical integration can be used for larger s.) The RSDs for muni and mnorm are again obtained from eq 2, but using the biased m estimates in place of µ. Accordingly, we can expect improved precision in m for µ > 2 but reduced precision at small µ. To correct mnorm to µ, we solve eq 7 for µ, obtaining

µ norm =

[

]

1/ 2 1 1 − (1 − 2mnorm s 2 ) , 2 s

(8)

Propagating the error from m to µ, we obtain

σµ =

σm

(1 − 2ms )

2 1/ 2

=

σm (1 − µs 2 ) ,

(9)

Huggett and coworkers13 modeled f(v) using the gamma distribution, which, when expressed in terms of the parameters α and β as

β α α −1 -βx Γ( x; α , β ) = x e , Γ(α )

(10)

has mean α/β and variance α/β 2, hence RSD α−1/2. Thus, for example, mean unity with 20% RSD can be achieved using α = β = 25. For estimating µ, the authors of Ref. 13 obtained  N  s  1 2   = 2 e ms − 1 ,  − 1    N 0   s   2

1 µ= 2 s

[


]

(11)


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which can be rearranged to give the biased estimate from naïve interpretation of the null count,

µ 2 s2 1 m = 2 ln(1+ µs 2 ) ≈ µ − , 2 s

(12)

where the approximation is obtained keeping just the lead terms in the expansion of the logarithm. This result is the same as eq 7 for the normal distribution. Using eq 11 to propagate the error in m into µ, we obtain

σ µ = (1 + µs 2 )σ m ,

(13)

which agrees with eq 9 in the supplement to Ref. 13. It may be noted that all uncertainties for m (and µ for monodisperse v) are based on eq 2 for binomial statistics. This result anticipates findings from the MC simulations, discussed next. Monte Carlo simulations Consider first the case of fixed v. The probability of a null for a specified µ is P0 = e−µ. The distribution of nulls and hits (positives) is binomial, so in Monte Carlo (MC) simulations, one can use a uniform random number generator for the default range 0-1 to designate the N partitions as nulls (RAN < P0) or hits. The MC statistics for nulls (N0) can be checked against the predictions of eq 2. For n simulations the standard error (SE) in N0 is sN0/n1/2; and for the SD in N0 computed from the n MC simulations, the relative SE is (2(n−1))−1/2 (from the properties of chi-square17). Accordingly, one expects the MC estimates to agree with predictions for N0 within its SE about 2/3 of the time; and similarly, the MC SD in N0 should deviate from its prediction (eq 2) by less than the factor (2(n−1)) −1/2 about 2/3 of the time. One can use this simple case to check the reliability of the random number generator. In this way I found some of the routines given by Press, et al.18 to be deficient for very small P0, settling on their RAN3 routine as satisfactory for these calculations. For variable v, the average copy number is λv. For simplicity, I use v models having average v = 1, giving average copy number µv and P0 = e−µν. A random number generator is used to pick volumes from the v distribution (e.g., the Box-Muller routine GASDEV in Ref. 18 for a normal v distribution), and a uniform random deviate is then used to designate each such partition as a null or


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a hit. To model the gamma distribution of Ref. 13, I have used the rejection method.18 I also have experimented with some 2-v models — 50:50 0.9 and 1.1, and 75:25 0.9 and 1.3 — both giving average v = 1, but with different symmetries. Like the case of fixed v, these models permit exact prediction (given earlier) that can be checked with the MC simulations. There are two ways of assembling statistics on µ and m from such simulations: (1) obtain the average N0 and its SD from the n simulations, each of N partitions, and convert these to m (and/or µ) and its SD; (2) estimate m (thence µ) from N0 for each simulation and then obtain the average and SD from these n estimates. The latter is consistent with experimental estimation, but results discussed below reveal that eq 1 is a biased estimator for µ, while N0 is unbiased. However, these biases are small, amounting to 1% for monodisperse v at µ = 8 (and less for polydisperse).

RESULTS AND DISCUSSION Monte Carlo All of the MC simulations have shown biases consistent with the predictions of eq 4 and precisions consistent with binomial statistics for m (µ for monodisperse v) from eqs 2 and 3. This means that when the RSDs in m are plotted vs m, they all fall on a single curve – the curve predicted

Figure 1. Relative standard deviation of estimated value m of average copy number µ for N = 104 partitions, as predicted for monodisperse v (solid line) and as estimated from MC simulations (n = 1000) for normally distributed v having 0%, 10%, and 20% RSD. The inset shows differences from predicted in the large-m region, with error bars indicating the MC precision.



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for µ with monodisperse v (Figure 1). On the other hand, when plotted vs the true µ, as obtained after correcting for the bias, the normal v models give slightly increased uncertainty for µ < 4, slightly increased precision for larger µ (Figure 2). These results bear out earlier predictions but are at odds with results shown in Fig. 8 of Ref. 11, in which even the results for monodisperse v do not agree with eq 2, which is equivalent to equations given for the RSD in µ in that work.

Figure 2. Comparison of MC RSDs in µ with predictions for monodisperse v and for normally distributed v with 20% dispersion.

For the 2-v model of eq 3, the negative bias is at most 5% at µ = 10. The SD exceeds that for monodisperse v by < 1% below µ = 4, and drops below it for larger µ. The MC results for the gamma distribution are fully consistent with the treatment of this model in the Supplement to Ref. 13. Equation 13 predicts loss of precision in the range 0.5 < µ < 3, as shown in this supplement; but it gives increased precision for µ > 4.5 (Figure 3). The maximum uncertainty increase is only 20% (µ ≈ 2), even for this extreme case of 50% dispersion. Figure 3 includes results for the biased estimate m; the difference in the two curves comes from correcting for the bias using eq 11. Extensive MC simulations were done for s = 0.5 (Figure 3) and for s = 0.2. Regarding the bias in eq 1 as an estimator of m or µ, the simulations reveal a 1% positive bias in µ at µ = 8 when the statistics for the n estimates of µ are processed for monodisperse v. This


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Figure 3. Bias (top) and relative standard deviation for partition volume v following the gamma distribution with mean 1 and RSD 0.5 (α = β = 4). In the lower plot the RSDs are ratioed to the values for monodisperse v, for the biased estimate m (eq 2) and for correction with eq 13.

bias drops by about half for the gamma distribution with s = 0.2 and by about half again for s = 0.5 (using eq 11 for µ). With monodisperse v, the bias drops to 0.03 at µ = 7. I can detect no bias in monodisperse estimates of µ based on the MC statistics for N0. Experimental treatment of partition volume dispersion Several groups have used optical microscopy methods to estimate partition volumes. In one extensive study of a droplet system, Pinheiro, et al.8 measured >1100 droplets from a total of16 wells for 5 different generator cartridges. Their Supplementary Figure S3 shows the 16 average volumes with standard deviations as error bars. The RMS SD is 0.0113 nL, which is 1.3% of the mean (0.868 nL) and only slightly larger than their estimated (Type B) 1% precision of measurement, suggesting little actual volume dispersion.19 These authors also claimed that these results showed no significant intra- or intercartridge dependence for the partition volume, but that assessment is incorrect. For example, a weighted average of the 16 means, using the SDs for



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weighting (as inverse variances) gives a χ2 values of 83, which is over 5 times the expected value of 15 (for 15 degrees of freedom), and more than twice the cut value for 0.001 probability of occurrence. But the situation is actually much worse, because the proper metric for comparing values is the SE, not the SD. Since each mean was based on ~70 measurements, the SEs are ~8 times smaller than the SDs, and the same weighted average now gives χ2 ≈ 6000. Even different wells for a given cartridge are mostly inconsistent. For example the first two volumes differ by 0.0135 nL which is 5 times their combined SEs. We can conclude that the overall SD of 0.026 (3%) is due to pseudorandom variability over the wells. Without the actual data for all droplets, it is not possible to assess their v distribution properly. However, assuming the values for each well are normally distributed, one can regenerate an approximation to all volumes by using the indicated means and SDs and a normal random error generator. The results of this exercise (Figure 4) show that the distribution is not Gaussian, and this remains true if the values for the anomalously low 9th well are omitted. However, it is likely that averaging over more wells would produce normally distributed vs, thanks to the Central Limit Theorem. In any event the 3% dispersion is small enough to produce negligible bias, and its effect on the estimation of λ can be accounted for adequately through simple error propagation. 200

150 count

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100

50

0 0.75

0.80

0.85 v (nL)

0.90

0.95

Figure 4. Histogrammed volumes from reconstruction of data for SI Figure S3 in ref 8. Error bars are taken as the square roots of the counts (Poisson approximation).20 The weighted LS fit to the normal distribution gives χ2 = 126, over 8 times the expected value for 15 degrees of freedom and confirming that the distribution is not normal.


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In an earlier study of a nanofluidic array system, Bhat, et al.4 found partition volume uncertainty to be a major part of their overall uncertainty of 6%, but it was not clear how much of their v uncertainty was true dispersion rather than measurement uncertainty. In more recent work by Corbisier, et al.14 on a droplet system, the emphasis was on the discovery of an 8% bias in the stated v for the system. The analysis was done in similar fashion to that of ref 8, by some of the same authors, and included 1794 droplets over 12 wells. The SD for single-well data was comparable to the earlier estimated 1%,8 but the overall diameter RSD ranged 1.2-2.7 %, giving 3 times larger RSD in the vs. As before, these RSDs seem attributable to interwell variability, but they remain too small to produce significant bias or loss of precision. Using a different droplet generation technology, Dangla et al.9 showed histograms of droplet radii for which the mean and spread both depended on operating pressure. The RSDs were stated to be less than 3%. While the data were displayed with superimposed Gaussian curves, the numbers of values were too small to make this convincing. Still, 3% RSD in the radii translates to ~9% in v, which is large enough to begin to show bias and precision effects of the sort considered here. It should be noted that if the radii are indeed normally distributed, the volumes will not be, thanks to their r3 dependence. For example, normally distributed radii with average 1.0 and SD = 0.1 give mean r3 = 1.028, SD = 0.304, and a distribution well approximated by the gamma distribution. An appealing alternative to optical characterization of partition volumes is calibration with reference materials, to which end much effort is currently being invested.15,16 From Figure 3, it is clear that any calibration of m vs µ must be nonlinear and weighted, but conducting such calibrations, including uncertainty estimation for the unknown, is simpler than is generally recognized.21,22 For example, if the v distribution is well approximated by the gamma distribution, then eq 12 represents a 1-parameter (s2) calibration relation, and one can obtain an estimate of the unknown and its SE from a 2-parameter nonlinear LS fit, as illustrated in Figure 5. If the calibration data deviate significantly from eq 12, it is easy to include extra calibration parameters in ad hoc fashion, with no added difficulty in extracting the SE for the unknown. And, if uncertainty in the calibrant µ values must be accommodated, that too is straightforward.23



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Figure 5. Hypothetical calibration plot21 for data following the gamma distribution (eq 12) with 50% dispersion, as in Figure 3. Calibration points are at µ = 0.1, 0.3, 1, 2, 4, and 8, and are weighted in accord with the binomial-based uncertainties in m at these points for N = 20,000. The single calibration parameter is a (s2), and b is the unknown µ, assigned an m value of 4 and the corresponding uncertainty. “Error” values are SEs.

CONCLUSION Surprisingly, polydispersity in digital PCR does not significantly increase imprecision, rather can actually improve precision for µ > 4. However, it is a source of bias, which can be considerable at large µ for large dispersion. Thus, it will be necessary to determine the distribution function in such cases. The optical microscopy methods discussed above have measurement precision of order 1% in the volume, so are capable of reliably doing such determinations for dispersion > 10%, where the bias and precision changes begin to become significant. Calibration with standard reference materials offers an indirect way to handle this problem, with errors in the knowledge of the distribution function being reflected in the calibration statistics. Then users can judge how much effort should go into such calibrations based on their needs. It is of course important at large µ that the reading techniques not miss any nulls, which, for example, number only ~7 out of 20,000 for monodisperse v at µ = 8.


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Below µ = 1, both the biases and precision losses from polydispersity are typically small enough to be acceptable without correction in many studies. Thus, as I have noted earlier, those who might design methods that can enhance throughput and reduce costs by allowing polydispersity should take inspiration from the present results. Further, although the present analysis has focused on dPCR, similar conclusions hold for any quantitative binary analysis method, where volume or sample size must be known and controlled, for example ELISA.

Acknowledgment This work was born at the recent qPCR and Digital PCR Congress held in Philadelphia (July, 2016). To those who may have been misled by my now-known-to-be incorrect comments after a presentation, predicting losses from polydisperity, I apologize. I have profited from exchanges with Bob Dorazio and Cody Youngbull, which were also initiated at this meeting.



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1992, 13, 444-449. (2) Higuchi, R.; Fockler, C.; Dollinger, G.; Watson, R. Biotechnology (N Y) 1993, 11, 1026– 1030. (3) Vogelstein, B.; Kinzler, K.W. Proc. Nat. Acad. Sci. USA 1999, 96, 9236-9241. (4) Bhat, S.; Herrmann, J.; Armishaw, P.; Corbisier, P.; Emslie, K. R. Anal. Bioanal. Chem. 2009, 394, 457-467. (5) Pekin, D.; Skhiri, Y.; Baret, J-C.; Le Corre, J.; Mazutis, L.; Ben Salem, C.; Millot, F.; El Harrak, A.; Hutchison, J. B.; Larson, J. W.; Link, D. R.; Laurent-Puig, P.; Griffiths, A. D.; Taly, V. Lab Chip 2011, 11, 2156-2166. (6) Kreutz, J. E.; Munson, T.; Huynh, T.; Shen, F.; Du, W.; Ismagilov, R. F. Anal. Chem. 2011, 83, 8158-8168. (7) Hindson, B.J.; Ness, K.D.; Masquelier, D.A.; Belgrader, P.; Heredia, N.J.; Makarewicz, A.J.; Bright, I.J.; Lucero, M.Y.; Hiddessen, A.L.; Legler, T.C. Anal. Chem. 2011, 83, 8604-8610. (8) Pinheiro, L. B.; Coleman, V. A.; Hindson, C. M.; Herrmann, J.; Hindson, B. J.; Bhat, S.; Emslie, K. R. Anal. Chem. 2012, 84, 1003-1011. (9) Dangla, R.; Kayi, S. C.; Baroud, C. N. Proc. Nat. Acad. Sci USA 2013, 110, 853-858. (10) Nixon, G.; Garson, J. A.; Grant, P.; Nastouli, E.; Foy, C. A.; Huggett, J. F. Anal. Chem. 2014, 86, 4387-4394. (11) Majumdar, N.; Wessel, T.; Marks, J.; PLoS One 2015, 10, e0118833. (12) Dorazio, R. M.; Hunter, M. E.; Anal. Chem. 2015, 87, 10886-10893. (13) Huggett, J. M.; Cowen, S.; Foy, C. A.; Clin. Chem. 2015, 61, 79-88. (14) Corbisier, P.; Pinheiro, L.; Mazoua, S.; Kortekaas, A.-M.; Chung, P. Y. J.; Gerganova, T.; Roebben, G.; Emmons, H.; Emslie, K. Anal. Bioanal. Chem. 2015, 407, 1831-1840. (15) Pavsic, J.; Zel, J.; Milavec, M. Anal. Bioanal. Chem. 2016, 408, 107-121.


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(16) Kline, M. C.; Romsos, E. L.; Duewer, D. L. Anal. Chem. 2016, 88, 2132-2139. (17) Tellinghuisen, J. Analyst 2008, 133, 161-166. (18) Press, W.H.; Flannery, B.P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes, Cambridge Univ. Press, Cambridge, U. K. (1986). (19) Data were not provided in the SI to ref 8, so were regenerated from Figure S3 using Web Plot Digitizer, http://arohatgi.info/WebPlotDigitizer/app/ (20) Tellinghuisen, J. J. Chem. Educ. 2005, 82, 157-166. (21) Tellinghuisen, J. Analyst 2005, 130, 370-378. (22) Tellinghuisen, J. Meth. Enzymol. 2009, 454, 259-285. (23) Tellinghuisen, J. Analyst 2010, 135, 1961-1969.



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(for TOC only)

0.07 Relative SD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

theory 0% v dispersion 10% 20%

0.05

0.03 6 7 Avg copy number

8


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Partition Volume Variability in Digital Polymerase Chain Reaction

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